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Classification Theory for Abstract Elementary Classes 3 Classification theory for abstract elementary classes Rami Grossberg ABSTRACT. In this paper some of the basics of classification theory for abstract elementary classes are discussed. Instead of working with types which are sets of formulas (in the first-order case) we deal instead with Galois types which are essentially orbits of automorphism groups acting on the structure. Some of the most basic results in classification theory for non elementary classes are presented. The motivating point of view is Shelah’s categoricity con- jecture for Lω1,ω. While only very basic theorems are proved, an effort is made to present number of different technologies: Flavors of weak diamond, models of weak set theories, and commutative diagrams. We focus in issues involving existence of Galois types, extensions of types and Galois-stability. Introduction In recent years the view that stability theory has wider applicability than the originally limited context (i.e. first-order stable theories) is getting increasing recognition among model theorists. The current interest in simple (first-order) the- ories and beyond signifies a shift in the opinion of many that similar tools and concepts to those of basic stability theory can be developed and are relevant in a wider context. Much of Shelah’s effort in model theory in the last 18-19 years is directed toward development of classification theory for non elementary classes.I feel that the study of classification theory for non elementary classes will not only provide us with a better understanding of classical stability (and simplicity) theory but also will develop new tools and concepts that will be useful in projecting new light on classical problems of “main stream” mathematics. The purpose of this article is to present some of the basics of classification theory for non elementary classes. For several reasons I will concentrate in what I consider to be the most important and challenging framework: Abstract Elemen- tary Classes, however this is not the only important framework for such a classifi- cation theory. 1991 Mathematics Subject Classification. Primary: 03C45, 03C52, 03C75. Secondary: 03C05, 03C55 and 03C95. I would like to express my gratitude to the anonymous referee for a most helpful report and many valuable comments. Also I am grateful to Yi Zhang, without his endless efforts this paper would not exist. 1 2 RAMI GROSSBERG I made an effort to keep most technical details to a minimum, while still having some deep ideas of the subject explained and part of the overall picture described. In the last section I discuss direction for future developments and some open prob- lems. An interested reader can find more details in Chapter 13 of [Gr]. I thank Oleg Belegradek for an interesting discussion and suggesting several examples, Andres´ Villaveces for his comments and Monica VanDieren for com- ments on a preliminary version. 1. definitions and examples DEFINITION 1.1. Let K be a class of structures all of the same similarity type L(K). K is an Elementary class if there exists a first-order T in L(K) such that K =Mod(T ). There are very many natural classes that are not elementary: Archimedean ordered fields, locally-finite groups, well-ordered sets, Noetherian rings and the class of algebraically closed fields with infinite transcendence degree. Extensions of predicate calculus permit a model-theoretic treatment of the above: 1. The basic infinitary languages: Lω1,ω L+,ω L+, and L∞,ω. The earliest work on infinitary logic was published in 1931 by Ernst Zermelo [Z]. Later pioneering work was done by: Novikoff (1939, 1943), Bochvar (1940) ([Bo]) and from the late forties to the sixties the main contributors were: Tarski, Hanf, Erdos,˝ Henkin, Chang, Scott, Karp, Lopez-Escobar, Morley, Makkai, Kueker and Keisler. Recall DEFINITION 1.2. Lω1,ω contains all the first-order formulas in the language L and is closed under propositional connectives, first-order quantification and the rule: If {ϕ (x) | n<ω}L then n _ ω1,ω ∈ ϕn(x) Lω1,ω,`(x) <ω. n<ω If {ϕ(x) | <}L+,ω then _ ϕ(x) ∈ L+,ω. < If {ϕ(x) | <}L+, with `(x) <then _ ϕ(x) ∈ L+, < CLASSIFICATION THEORY FOR ABSTRACT ELEMENTARY CLASSES 3 and for every sequence of variables hx | <<i Qx0Qx1 Qx ϕ(hx | <i) ∈ L+, for Q ∈{∀, ∃}. For a limit cardinal : [ L, := L+,. < L∞, is the proper class obtained by letting vary over all cardinals. I.e. it is [ L∞, := L+,. ∈Card Basic references: For Lω1,ω see H. J. Keisler [Ke2] and for L+, see M. A. Dickmann’s [Di] books. 2. Cardinality quantifiers: Andrzej Mostowski in 1957 (see [Mos]) introduced several cardinality quantifiers. The most popular among them was studied extensively in the sixties by Gerhard Fuhrken [Fur] and Jerry Keisler [Ke1]. ℵ It is the 1-interpretation: L(Q) and extensions like Lω1,ω(Q) where M |= Qxϕ(x) ⇐⇒ { a ∈|M| : M |= ϕ[a]} is uncountable. 3. Other second order quantifiers: Measure-theoretic quantifiers, and topolog- ical logics. Barwise, Makkai, Kaufmann, Flum, Ebbinghaus and Ziegler. 4. Other logics: Monadic logics. Rabin, Gurevich, Buchi,¨ Shelah. There are many more logics, see the volume edited by Jon Barwise and Solomon Feferman [BaFe]. There is a particularly rich model theory for L(Q). In the last 20 years, this theory took on a set-theoretic flavor, see: Fuhrken [Fur], Keisler [Ke1], [Sh 82], [Gr2] and [HLSh]. For an overview of some of this I recommend Wilfrid Hodges’s book [Ho1]. A common feature of all the above extensions of first order logic is the failure of the compactness theorem. An ordered field hF, +, , i is archimedean iff _ hF, +, , i |= ∀x x>0 → [|x + ...{z + x} 1] . n<ω n-times A group G is periodic iff _ G |= ∀x [xn =1]. n<ω DEFINITION 1.3. Let M and N both be L-structures. Suppose that A |M|, and B |N|. A function f : A → B is called a partial isomorphism iffitisa bijection and for every relation symbol R(x) and every a ∈ A we have that a ∈ RM ⇐⇒ f(a) ∈ RN , for every function symbol F (x) we have that 4 RAMI GROSSBERG f(F M (a)) = F N (f(a)) and for every constant symbol c if cM ∈ Dom(f) then f(cM )=cN . A family F of partial isomorphisms from M into N has the back and forth property iff (forward) for every a ∈|M| and every g ∈Fthere exists h ∈Fsuch that g h and a ∈ Dom(h) and (back) for every b ∈|N| for every g ∈Fthere exists h ∈Fsuch that g h and b ∈ Rang(h). Denote this by F : M =p N . We write M =p N for “there exists a non empty F such that F : M =p N”. At first glance the following special case of a theorem of Carol Karp may look a bit surprising: FACT 1.4 (Karp’s test). Let M and N be L-structures. p M = N ⇐⇒ M ∞,ω N. Shelah’s plenary talk at the International Congress of Mathematicians in Berke- ley in 1986 (see [Sh 299] and [Sh tape]) was dedicated to classification theory for non elementary classes and in particular universal classes. There are several frameworks for classification theory for non elementary classes. With the exception of the last item they are listed in (more or less) increasing level of generality: Homogeneous model theory (this was formerly called Finite diagrams sta- ble in power). In this context we have a fixed (sequentially) homogeneous monster model M and limit the study to elementary submodels of M and its subsets. The subject was started by Shelah in 1970 with [Sh3] and con- tinued by him with [Sh54]. In the last 10-15 years the subject was revived in several publications. See [Gr3], [Gr4],[HySh1], [HySh2], [GrLe1], [GrLe3], [Le1], [Le2], [BuLe], [Be] and [Kov1]. Submodels of a given structure. This is a generalization of homogeneous model theory. Start with a given model M (not necessarily homogeneous) and limit the study to submodels of M and its subsets. Started in Grossberg [Gr3], [Gr4]. Further progress is in [GrLe2]. Excellent classes. An excellent class consists of the atomic models of a first- order countable theory which satisfy a very strong amalgamation property: The (ℵ0,n)-goodness for every n<ω. Shelah introduced them in [Sh87a], ∈ [Sh87b] as a tool to analyze Mod() when Lω1,ω under the model the- ℵ oretic assumption that I(ℵn,) < 2 n holds for every positive integer n. Later in [Sh c] Shelah used excellent classes at a crucial point in his proof of the main gap for first-order theories, (ℵ0,n)-goodness is renamed in [Sh c], page 616 as (ℵ0,n)-existence property. In section 5 of chapter XII She- lah essentially show that for countable and superstable T without the dop, notop is equivalent to the (ℵ0, 2)-existence property which using his resolu- tion technique from [Sh87b] implies excellence. Lately Kolesnikov [Kov2] announced an n-dimensional analysis to get new results for simple unstable CLASSIFICATION THEORY FOR ABSTRACT ELEMENTARY CLASSES 5 theories. Grossberg and Hart in [GrHa] developed orthogonality calculus to the level that permitted deriving the main gap for excellent classes. This was the first time that a main gap was proved for non elementary classes. Universal classes. TheV prototypical example is Mod() when is an Lω1,ω sentence of the form n<ω n where the n are 1-first-order sentences. This work began in [Sh 300]. Shelah is currently writing a book [Sh h] that, among other things, will include a “main gap”-style of theorem for universal classes. Abstract elementary classes.
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